Group #142069 — subgroups

Subgroups — #142069

Order 512.

289 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33296	C₂	1	1	yes	maximal, characteristic
2	#12658	C₂	1	1	yes	maximal, characteristic
3	C₄×#347	C₂	1	1	yes	maximal, characteristic
4	#7266	C₂	1	4	yes	maximal
5	#2862	C₄	1	1	yes	characteristic
6	#2859	C₄	1	1	yes	characteristic
7	#2150	—	2	1	no	—
8	C₂×#347	—	2	1	no	—
9	C₂×#347	C₂²	1	2	yes	—
10	C₂×#347	C₂²	1	2	yes	—
11	C₄×M₃₂	—	2	1	no	—
12	#1993	—	2	1	no	—
13	#1993	—	2	1	no	—
14	#1991	—	2	1	no	—
15	#1991	—	2	1	no	—
16	#1836	C₄	1	2	yes	—
17	#1836	C₂²	1	2	yes	—
18	C₄×#97	C₂²	1	1	yes	characteristic
19	#1375	—	2	1	no	—
20	C₂×C₄⋊Q₈	—	2	1	no	—
21	C₂×#125	—	2	1	no	—
22	C₂×M₃₂	—	2	2	no	—
23	C₂×M₃₂	—	2	2	no	—
24	C₂×#97	C₂×C₄	1	1	yes	characteristic
25	C₂×#97	D₄	1	1	yes	characteristic
26	C₂×#97	D₄	1	1	yes	characteristic
27	C₂×#97	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	C₂×C₄×C₈	D₄	1	1	yes	characteristic
30	#391	—	4	2	no	—
31	#389	—	4	2	no	—
32	#384	C₂×C₄	1	2	yes	—
33	#382	—	2	1	no	—
34	#382	—	2	1	no	—
35	#382	C₂×C₄	1	2	yes	—
36	C₄³	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₁₆	—	2	4	no	—
38	#347	—	2	4	no	—
39	#347	—	2	2	no	—
40	#347	—	2	4	no	—
41	#347	Q₈	1	4	yes	complemented
42	#346	—	2	2	no	—
43	C₄×C₁₆	—	2	2	no	—
44	#340	—	2	2	no	—
45	#340	—	2	2	no	—
46	#340	—	2	2	no	—
47	#340	—	2	2	no	—
48	C₂²×Q₈	—	4	1	no	—
49	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
53	C₄⋊Q₈	—	4	2	no	—
54	C₄⋊Q₈	—	4	2	no	—
55	#125	—	4	2	no	—
56	#125	—	4	1	no	—
57	#125	—	4	1	no	—
58	C₂×C₄⋊C₄	—	2	2	no	—
59	C₂×C₄⋊C₄	—	2	2	no	—
60	C₂×C₄⋊C₄	—	4	2	no	—
61	C₂×C₄⋊C₄	—	4	1	no	—
62	C₂×C₄⋊C₄	—	4	2	no	—
63	C₂×C₄²	—	2	2	no	—
64	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
67	M₃₂	—	2	4	no	—
68	M₃₂	—	2	4	no	—
69	M₃₂	—	4	2	no	—
70	M₃₂	—	4	2	no	—
71	C₂×C₁₆	—	2	4	no	—
72	C₂×C₁₆	—	2	4	no	—
73	#97	—	2	2	no	—
74	#97	—	2	2	no	—
75	#97	—	4	1	no	—
76	#97	C₂×Q₈	1	2	yes	—
77	#97	#41	1	2	yes	—
78	C₄×C₈	—	2	1	no	—
79	C₄×C₈	—	2	1	no	—
80	C₄×C₈	—	2	2	no	—
81	#95	—	4	2	no	—
82	#95	—	4	2	no	—
83	#95	—	4	2	no	—
84	C₂×Q₈	—	8	2	no	—
85	C₂×Q₈	—	8	4	no	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	—	4	2	no	—
90	C₂²×C₄	#100	1	2	yes	—
91	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	#101	1	2	yes	—
102	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₈	#124	1	1	yes	characteristic
104	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	8	2	no	—
110	C₄⋊C₄	—	8	4	no	—
111	C₄⋊C₄	—	8	2	no	—
112	C₄⋊C₄	—	8	2	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	8	2	no	—
116	C₄²	#104	1	2	yes	—
117	C₄²	#104	1	2	yes	—
118	C₄²	#104	1	1	yes	characteristic
119	C₄²	#104	1	1	yes	characteristic
120	C₄²	C₄×Q₈	1	2	yes	—
121	C₁₆	—	2	8	no	—
122	C₁₆	—	4	4	no	—
123	C₂³	C₂×#100	1	1	yes	characteristic
124	Q₈	—	16	4	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	—	8	4	no	—
132	C₂×C₄	—	8	2	no	—
133	C₂×C₄	—	8	2	no	—
134	C₂×C₄	#329	1	2	yes	—
135	C₂×C₄	#329	1	2	yes	—
136	C₂×C₄	#391	1	1	yes	characteristic
137	C₂×C₄	C₂×#101	1	1	yes	characteristic
138	C₂×C₄	C₂×#104	1	1	yes	characteristic
139	C₂×C₄	C₂×#104	1	1	yes	characteristic
140	C₂×C₄	C₂×#104	1	1	yes	characteristic
141	C₂×C₄	#421	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	4	1	no	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂×#329	1	1	yes	characteristic
149	C₂²	#1527	1	1	yes	characteristic
150	C₂²	#1997	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	4	2	no	—
155	C₄	—	16	2	no	—
156	C₄	—	16	2	no	—
157	C₄	#1526	1	1	yes	characteristic
158	C₄	#1531	1	1	yes	characteristic
159	C₄	#1986	1	1	yes	characteristic
160	C₄	#1994	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	#9890	1	1	yes	characteristic
164	C₂	#11911	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33296

C₂

yes

maximal, characteristic

#12658

C₂

yes

maximal, characteristic

C₄×#347

C₂

yes

maximal, characteristic

#7266

C₂

yes

maximal

#2862

C₄

yes

characteristic

#2859

C₄

yes

—

—

C₂²

yes

—

C₂×#347

C₂²

yes

—

C₄×M₃₂

—

—

—

—

—

C₄

yes

—

#1836

C₂²

yes

—

C₄×#97

C₂²

yes

characteristic

#1375

—

C₂×C₄⋊Q₈

—

C₂×#125

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

C₂³

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

—

C₂×C₄

yes

—

#382

—

#382

—

#382

C₂×C₄

yes

—

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

—

—

—

Q₈

yes

complemented

#346

—

C₄×C₁₆

—

—

—

—

—

C₂²×Q₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

C₂×Q₈

yes

—

#97

#41

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

#95

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#100

yes

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

#101

yes

—

102

C₂×C₈

C₂²⋊Q₈

yes

—

103

C₂×C₈

#124

yes

characteristic

104

C₂×C₈

C₄⋊Q₈

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

#104

yes

—

117

C₄²

#104

yes

—

118

C₄²

#104

yes

characteristic

119

C₄²

#104

yes

characteristic

120

C₄²

C₄×Q₈

yes

—

121

C₁₆

—

122

C₁₆

—

123

C₂³

C₂×#100

yes

characteristic

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#329

yes

—

135

C₂×C₄

#329

yes

—

136

C₂×C₄

#391

yes

characteristic

137

C₂×C₄

C₂×#101

yes

characteristic

138

C₂×C₄

C₂×#104

yes

characteristic

139

C₂×C₄

C₂×#104

yes

characteristic

140

C₂×C₄

C₂×#104

yes

characteristic

141

C₂×C₄

#421

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₂²

—

148

C₂²

C₂×#329

yes

characteristic

149

C₂²

#1527

yes

characteristic

150

C₂²

#1997

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#1526

yes

characteristic

158

C₄

#1531

yes

characteristic

159

C₄

#1986

yes

characteristic

160

C₄

#1994

yes

characteristic

161

C₂

—

162

C₂

—

163

C₂

#9890

yes

characteristic

164

C₂

#11911

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#329
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#329

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₂×#97

C₂³

prime radical

C₂³

C₂×#100

LCS

G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G

LECS

G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
41. #347	Q₈	semidirect

H ≅ G/N

type

41. #347

Q₈

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#33296	2
#12658	2
C₄×#347	2
#7266	2

subgroup

index

Subgroups — #142069

Conjugacy classes of subgroups

Complements

Maximal subgroups